LAB 01 Experimental Errors and Data Analysis

Transcription

1 LAB 01 Experimental Errors and Data Analysis COTET: 1. Introduction 2. Experimental Issues 3. Experimental Errors A. Classification B. Calculations C. Reporting 4. Pre-lab: A. Activities B. Preliminary info C. Quiz It is scientific only to say what is more likely and what less likely, and not to be proving all the time the possible and impossible. Richard Feynman 1. Introduction As we emphasized many times in class, Physics is a natural science driven by scientific method, which is a standardized process designed to build a model of reality devoid of subjective interferences, such as cultural prejudices and biases. In its standard form, the scientific method employs three steps 1. Observation focused on certain aspects of a natural phenomenon. 2. Formulation of a hypothesis deemed to explain the phenomenon tentatively via a descriptive or mathematical causal mechanism. To be acceptable as a scientific hypothesis, the putative mechanism must be falsifiable experimentally: that is, it must be capable to make testable predictions about the evolution of the respective phenomenon. 3. Experimental validation of the predictions in a reproducible manner by independent experimenters. Once the predictions are tested, the hypotheses can be elevated to the status of scientific theory acceptable within the limits of the experimental setup used to verify it. It is important to understand that science abhors hubris and it remains a healthy endeavor only as long as it is practiced with a creative dose of skepticism. It is a common occurrence in the history of science that mighty long-standing theories about natural phenomena to be eventually disproved or reduced to merely particular cases of more general models. For example, look no further than to ewtonian mechanics (about to be introduced as an archetypal science this semester) which dominated natural sciences for two centuries before it was dethroned to the status of a macroscopic low-speed limit of quantum mechanics and relativistic mechanics. ote that there is nothing esoteric about the scientific method. You can apply it to otherwise prosaic situations such as fixing a problem with your car. The first epiphany in one s quest to understanding science ought to be realizing the effective simplicity of its basic methods. To frame properly the subject of our first PHYS-14 lab, and of this lab class in general, let us take a closer look at the experimental testing which reigns supreme in validating hypotheses. 2. Experimental Issues As highlighted above, experiments cannot confirm anything ultimately. In fact, it is important to note that the purpose of the scientific method is not to prove hypotheses correct, but tentatively not false, because experimental results are valid only within a limited range of measurable parameters which can be used at most to define a domain of applicability rather than justify any pretensions of universality. Incidentally, this nuances the example above with ewtonian mechanics: we see that its birth, longevity and popularity is hardly fortuitous inasmuch as the domain of applicability of its models overlaps with the domain of our everyday experience. Thence, in Physics we talk about the classical limit which sets the adequacy of ewtonian mechanics only for objects much larger and more massive than particles of atomic scale, and moving much slower than light in vacuum. How come experiments are inherently limited to certain domains of relevance? And how do we actually measure and quantify the uncertainty associated with a particular experimental setup? In brief, experimental relevance is limited both by apparatus and the intrinsic nature of measured objects. Reality has blurry edges ; it actually comes to appear fairly well defined only due to the macroscopic coherence arising statistically from an otherwise chaotic microscopic structure. 1

2 To quantify and put in critical perspective this uncertainty of particular experiments, we use error analysis performed on experimental data. Organizing and conducting an experiment can be a rather complex job, demanding that one applies ingenuity to very particular physical situations. Yet, many experiments follow a common strategy where a quantity call it an independent variable is changed whilst another quantity call it a dependent variable is monitored; meanwhile other factors call them control variables are kept constant. Thence, inasmuch as the two types of variables are related via the very mechanism that is being tested, the experiment will serves as a validation machine for the respective hypothesis. Example 1: Imagine that you want to verify an idea that you will hear in this class: considering an object sliding on a slippery surface, in a first approximation the contact friction will depend only on the nature of the two surfaces in contact and the weight of the object. This is a somewhat disconcerting statement. For once, it implies that the friction is independent of how wide is the contact area. So, in a fit of critical thinking, you may demote the idea to the status of an unverified hypothesis, and perform an experiment to confirm it using a block of wood with faces of different areas. The block could be launched on a long table to see if the contact area has an impact on friction. In this case, the independent variable would be the face of the object, the dependent variable could be the stopping distance (a measure of the strength of friction), while a few other parameters would be control variables, such as the weight and the initial speed of the block. The hypothesis will be confirmed if the stopping distance remains the same irrespective of the sliding face. ote that the relevance of an experiment may be affected by other factors rather than objective inherent errors. One such factor is dishonesty. Another is the unconscious or negligent cherry-picking of data to prop the preference or bias of the researcher for a certain hypothesis. However, this is not the kind of errors that we shall be studying in this lab. We shall graciously assume honesty and responsibility in our data collection, analysis and reporting. So let us focus on the unavoidable errors associated with experimental setups. 3. Experimental Errors A. Classification For completion and scientific respectability, the errors affecting an experiment must be specified. ot only that they are unavoidable, but they are technically relevant as part of the clockwork of data analysis. Additionally, in an instructional setup like ours where we merely test well established models, errors can be used to compare the performance of different lab teams, and to check the results against published data. Experimental errors are typically classified into systematic (determinate) and random (indeterminate or statistical): Systematic errors are typically due to faulty instruments or measuring techniques affecting the measurement in only one direction by the same deviation. Random errors are due to the inherent variations in the measured characteristics due to uncontrollable causes such as thermal fluctuations, tiny vibrations, or microscopic irregularities. The impact of these erratic variations can be reduced, but never completely eliminated. Therefore, repeated measurements of the same quantity in the same setup will result in numbers that will be distributed on both sides about an average value within a range proportional to the amplitude of the error. Thus, the spread of the data about this average is a measure of the uncertainty of the measurement. In lab parlance, the accuracy and precision of a measurement are different concepts as they are determined by different types of experimental errors. Thus, accuracy is a measure of how much the measured value deviates from an expected value, such that a low accuracy is likely determined by a systematic error. In turn, precision is related to the agreement of repeated experiments affected by the uncertainty of the experimental setup indicated by the data spread about the expected value. Therefore, a repeated measurement with low accuracy but high precision will result in data values tightly grouped about an average showing a large deviation from the expected value. Failing to account carefully for the sources and types of error can hinder the correct interpretation of experimental results. For instance, data features produced by systematic errors may be confused for physical phenomena, whilst legitimated discoveries may be conversely attributed to systematic errors. This is why the reproducibility of experimental results by different research groups is a highly priced validation criterion and practice in science. 2

3 B. Calculations As we have noticed above, a respectable experimental scientific report ought to specify errors. This can be done either numerically or using error bars on graphical renderings of data. There are quite a few formulas that can be employed to capture the uncertainty of experimental measurements. Percent Error: An experiment can be used to estimate numerically a quantity and then compare it to a putatively precise expected value. The expected value may be a generally accepted number (like a universal constant), or may be calculated using a mathematical expression, or may be the average (or mean) of repeated measurements x 1,2,3 of the quantity: 1 x x1 x2 x3... x (1) In this case, denoting X the expected value and x the experimental result, the adequacy of the measurement can be quantified using the percent error: x X PE 100%. (2) X Mean Deviation: Finding the actual sources of random errors polluting the data of an experiment may be a rather hopeless task, but their impact can be fairly easy to quantify owing to their random nature. Thus, when measurements are repeated in identical conditions, the data x 1,2,3 tend to disperse about the mean value x. Then, the uncertainty can be characterized by calculating the mean deviation: x x x x... x x MD x x. (3) i i1 Standard Deviation: An even better measure of experimental uncertainty is the so called standard deviation which when based on sufficient repeats estimates the maximum spread of values about the mean: C. Reporting x1 x x2 x... x x i. (4) 1 SD x x i1 Example 2: Suppose that, in the friction experiment described earlier, you measured the stopping distance travelled by the sliding block times for each face. The data are tabulated on the right. Then, the mean value of the distance is x cm Thence, using the proper significant figures, the mean deviation and standard deviation are about the same: MD cm SD cm Once the mean deviation or standard deviation is calculated, it is customary to report the result x of the respective measurement in one of the following forms which indicate the precision of the respective measurement using one of the deviations about the mean: x x MD or x x SD. () The deviation may be also reported as a percent of the mean (relative error): MD SD xx 100% or xx 100%. (6) x x Therefore, the various forms the distance can be reported either as 41 ± 2 cm or 41 cm ± %. To verify the hypothesis, the distances measured on the other two faces of the block should be comparable within an error of about 2 centimeters. Trial Distance (cm)

4 Most Physics articles report experimental findings using graphs, inasmuch as plotting data can prove useful not only for reporting per se, but also as a heuristic tool. Accordingly, graphical representations will be ubiquitous in this lab and you will be kindly required to review a modicum of basic rules for data plotting: 1. Choose axis scales consistent with the scale of the represented data sets, such that the graph will be commensurate with the size of the frame. (For instance, a spatial axis shouldn t be scaled in tens of centimeters if the represented data is in millimeters; the graph will look too small.) 2. Title the plot, label the axes meaningfully, specify units, and use a legend if the graph includes more than one curve. 3. After the data points are plotted, draw a smooth line indicating the trend of the data distribution. That is, avoid connecting point with the line: recall that the data points deviate from their mean with equal probability on each side, and the line should be indicative for the mean. So the line should be a curve of best fit." 4. The errors can be represented on the graph using error bars with length given by the deviation associated with each point. MD or SD MD or SD 4. Pre-lab A. Activities 1. Peruse carefully through the introductory material provided above. In case a certain concept is not clear enough, feel free to contact me or your lab-instructor (if different) with any question that you may have. Make sure you understand the two types of errors that can occur in measuring and how to compensate or correct for each type of error. Compare and understand the differences between accuracy and precision. Make sure you understand how to compute the three types of errors presented in Subsection 3B. You will be using these tools to analyze your data which will help when formulating your conclusions. 2. Read the preliminary information provided below and riffle through the associated Lab Form to familiarize yourself with the theoretical background of the experiment and the lab tasks that you and your team are to eke through during the next lab session. 4. Answer the questions on the quiz at the end of this document. The questions are also available on the Blackboard site associated with the PHYS-14 lecture (not lab), and it is strongly recommended to use that system. However, in special situations such as when you don t have access to internet when needed, you may answer the questions on paper and turn them in at the beginning of the lab class.. If possible, bring your laptop to lab next week. Incorrect Correct B. Preliminary information The subject of our first lab is Experimental Errors and Data Analysis. It will give you an opportunity to practice some of the concepts introduced above. To that end, we shall imagine and analyze a concrete physical situation presented as a scenario (see the Lab Form) involving people swinging out on a rope. You will be required to come with a hypothesis and then test it regarding the dependence of the swinging time on the weight of the dangling person. To analyze the scenario, you will set up an experiment using a simple pendulum meant to model the people swinging out at the end of a rope. ote that the subject of the experiment is not the pendulum per se, but the method of data analysis, so you don t have to worry about the theory behind the behavior of this device. 4

5 Pendulum Concepts: The time it takes for the pendulum to complete a swing back and forth is called the Period (T). The distance from the pivot point to the center of mass of the pendulum bob is the Arm length (L) The displacement relative to the vertical can be obtained either by measuring the deflection angle () or the height (h) above the lowest point the bob passes through. These can be related through right triangle trigonometry via h L 1 cos. (7) A theoretical value for the time period T of a simple pendulum can be approximated using the following formula valid when the displacement angle is small (less than or equal to 20 ): h bolt bob trajectory string/rod massive bob L T 2, (8) g where g is the gravitational acceleration, g = 9.8 m/s 2. ote that, according to this formula, the mass of the swinging people shouldn t influence the period as long as the displacement angle is small. (Quiz on the next page)

6 C. Quiz 1 ame: Based on the pre-lab readings, please answer the following questions on the Blackboard site associated with our PHYS 14 lecture. The numbers in square brackets indicate the points allotted for the respective question. Q1. [2] In scientific research, a theory is expected to be tested experimentally by more than one research group because a) this may eliminate the impact of hidden systematic errors. b) this may eliminate the impact of random errors. c) to be acceptable, experimental results must be reproducible. d) scientific theories lose validity in time, so they must be reconfirmed periodically. e) Both (a) and (c) are true. Q2. [2] One of the earliest observations of modern physics was that falling objects will accelerate at the same rate provided that only their weight acts on them. One can immediately conceive an experiment to test this idea: just drop objects of different mass from the same height and verify that they hit the ground at the same time. In this experiment, what kind of variables are the mass, the falling time, and the height, in this order? a) dependent, independent, control b) control, independent, dependent c) independent, dependent, control d) independent, control, dependent e) systematic, random, standard Q3. [2] In the experiment depicted above, what kind of error would be introduced by air resistance? a) random b) systematic c) independent d) standard e) mean Q4. [2] Consider the following data set representing a quantity measured repeatedly 4 times: {3.2, 3.1, 3.3, 3.2}. Say that, based on a theoretical model, one calculated the expected value for the respective quantity to be 3.2. What are the percent error of the set average relative to the expected value, and the mean deviation of the set? a) 0.0 and 0 b) 0 and 3.2 c) 3.2 and 0.0 d) 0 and 0.0 e) one of the above. Q. [2] Based on expression 8 in the Pre-Lab, which of the following will not influence the period of a pendulum at arbitrary swinging angles? a) the arm length. b) the planet where the period is measured. c) the mass of the pendulum. d) the displacement angle when small. e) Actually, all of the above will influence the period. 6